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Nonlocal Flow Driven by the Radius of Curvature with Fixed Curvature Integral

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Abstract

This paper deals with a \(1/\kappa \)-type nonlocal flow for an initial convex closed curve \(\gamma _{0}\subset {\mathbb {R}}^{2}\) which preserves the convexity and the integral\(\ \int _{X\left( \cdot ,t\right) }\kappa ^{\alpha +1}ds,\ \alpha \in \left( -\infty ,\infty \right) ,\) of the evolving curve \(X\left( \cdot ,t\right) \). For\(\ \alpha \in [1,\infty ),\ \)it is proved that this flow exists for all time \(t\in [0,\infty )\) and \(X(\cdot ,t)\) converges to a round circle in \(C^{\infty }\) norm as \(t\rightarrow \infty \). For \(\alpha \in \left( -\infty ,1\right) \), a discussion on the possible asymptotic behavior of the flow is also given.

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Notes

  1. In this paper, we will always assume the constant \(\alpha \) to be nonzero. We will not write ”\(\alpha \ne 0\)” from now on. If \(\alpha =0,\ \)then the integral quantity \(E\ \)in (5) is the constant \(2\pi ,\) which is automatically preserved and there is nothing to be discussed.

  2. In [12], the linear nonlocal PDE is decomposed into the linear heat equation plus certain simple ODE (see line 6 in p. 240 of [12]).

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Acknowledgements

We are deeply grateful for the reviewer’s careful reading, comments, and suggestion on our paper. Laiyuan Gao is supported by the National Natural Science Foundation of China (No. 11801230). Shengliang Pan is supported by the National Natural Science Foundation of China (No. 11671298) and by the Science Research Project of Shanghai (No. 16ZR1439200). Dong-Ho Tsai is supported by the MoST of Taiwan with Grant No. 105-2115-M-007-007-MY3.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Jiangsu Normal University, No.101, Shanghai Road, Xuzhou, Jiangsu, China

    Laiyuan Gao

  2. School of Mathematical Sciences, Tongji University, No.1239, Siping Road, Shanghai, China

    Shengliang Pan

  3. Department of Mathematics, National Tsing Hua University, Hsinchu, 30013, Taiwan

    Dong-Ho Tsai

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  1. Laiyuan Gao
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  2. Shengliang Pan
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  3. Dong-Ho Tsai
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Correspondence to Dong-Ho Tsai.

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Gao, L., Pan, S. & Tsai, DH. Nonlocal Flow Driven by the Radius of Curvature with Fixed Curvature Integral. J Geom Anal 30, 2939–2973 (2020). https://doi.org/10.1007/s12220-019-00185-4

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